If N is a large, positive integer, Stirling's approximation states that N! is approximately given by

We can show that this is approximately correct by using the relation between a sum and an integral. Since ln x is a monotonic increasing function of x, then

These inequalities can be made obvious by graphing the function ln x and comparing it to the summed areas of the unit stepwise divisions representing the sums in (A.3.2). From (A.3.2) it follows that

Performing the integrals, and recognizing that the sum in the middle is ln N!, we get

If unity is neglected relative to N, equation (A.3.1) follows immediately. It is trivial to show that the outside terms in (A.3.4) differ by a quantity of order ln N. For very large numbers, the logarithm is much smaller than the number, so the greater N, the more accurate is (A.3.1).

The following table shows that Stirling's approximation is a good one for remarkably small values of N:

However, there are times when (A.3.1) is not sufficiently accurate even for large values of N because of cancellations occurring in ratios of factorials. It is then necessary to use the following more accurate formula:

This approximation neglects terms of order N ^?3 and...